Parseval Frame Research Articles

Continuous frames in Krein spaces

The purpose of this paper is to propose a definition of continuous frames of rank n for Krein spaces and to study their basic properties. Similarly to the Hilbert space case, continuous frames are characterized by the analysis, the pre-frame and the frame operator, where the latter gives rise to a frame decomposition theorem. The paper includes a discussion of similar, dual and Parseval frames and of reproducing kernels. In addition, the importance of the fundamental symmetry in the formula for the frame operator in a Krein space is clarified. As prime examples, it is shown how to transfer continuous frames for Hilbert spaces to Krein spaces arising from a possibly non-regular Gram operator.

Directional Analytic Discrete Cosine Frames

Block frames called <i>directional analytic discrete cosine frames</i> (DADCFs) are proposed for sparse image representation. In contrast to conventional overlapped frames, the proposed DADCFs require a reduced amount of 1) computational complexity, 2) memory usage, and 3) global memory access. These characteristics are strongly required for current high-resolution image processing. Specifically, we propose two DADCFs based on discrete cosine transform (DCT) and discrete sine transform (DST). The first DADCF is constructed from parallel separable transforms of DCT and DST, where the DST is permuted by row. The second DADCF is also designed based on DCT and DST, while the DST is customized to have no DC leakage property which is a desirable property for image processing. Both DADCFs have rich directional selectivity with slightly different characteristics each other and they can be implemented as non-overlapping block-based transforms, especially they form Parseval block frames with low transform redundancy. We perform experiments to evaluate our DADCFs and compare them with conventional directional block transforms in image recovery.

On description of dual frames

One of key problems in signal reconstruction process with the use of frames is to find a dual frame. Typically, a canonical dual frame is used. However, there are many applications where this choice appears to be unfortunate. Due to that fact, it is necessary to develop a tool, which helps to find a suitable dual frame. In this paper we give a method to find every dual frames. The proposed method is based on Naimark's dilation theorem and the obtained description of dual frames involves parameters that characterize extension of a Parseval frame to an orthonormal basis. These formulas are simplified for frames in finite-dimensional spaces and for near-Riesz bases. In the latter case, the simplification is based on the extended and supplemented version of the Naimark theorem, which is proved in the last part of the paper.

Parseval Wavelet Frames on Riemannian Manifold

We construct Parseval wavelet frames in $$L^2(M)$$ for a general Riemannian manifold M and we show the existence of wavelet unconditional frames in $$L^p(M)$$ for $$1< p <\infty $$ . This is made possible thanks to smooth orthogonal projection decomposition of the identity operator on $$L^2(M)$$ , which was recently proven by Bownik et al. (Potential Anal 54:41–94, 2021). We also show a characterization of Triebel–Lizorkin $${\mathbf {F}}_{p,q}^s(M)$$ and Besov $${\mathbf {B}}_{p,q}^s(M)$$ spaces on compact manifolds in terms of magnitudes of coefficients of Parseval wavelet frames. We achieve this by showing that Hestenes operators are bounded on $${\mathbf {F}}_{p,q}^s(M)$$ and $${\mathbf {B}}_{p,q}^s(M)$$ spaces on manifolds M with bounded geometry.

The Paulsen problem made simple

The Paulsen problem was once thought to be one of the most intractable problems in frame theory. It was resolved in a recent tour-de-force work of Kwok, Lau, Lee and Ramachandran. In particular, they showed that every ϵ-nearly equal norm Parseval frame in d dimensions is within squared distance O(ϵd13/2) of an equal norm Parseval frame. We give a dramatically simpler proof based on the notion of radial isotropic position, and along the way show an improved bound of O(ϵd2).

A new probabilistic model for optimal frames in erasure's recovery

In this paper we introduce a new probabilistic model for the occurrence of erasures in multiple channel data transmission. It uses Parseval frames to encode the information to be transmitted. Information loss in the transmission channels is modeled by a sequence of Bernoulli random variables. Our model gives insights on the probabilistic properties of the channels and allows us to find optimal Parseval frames that minimize the lost of data in the worst one erasure case. We show also that compared to existing models [3,4,7,8], our optimal Parseval frames give better performance for recovering transmitted data.

Signal Processing on the Permutahedron: Tight Spectral Frames for Ranked Data Analysis

Ranked data sets, where m judges/voters specify a preference ranking of n objects/candidates, are increasingly prevalent in contexts such as political elections, computer vision, recommender systems, and bioinformatics. The vote counts for each ranking can be viewed as an n! data vector lying on the permutahedron, which is a Cayley graph of the symmetric group with vertices labeled by permutations and an edge when two permutations differ by an adjacent transposition. Leveraging combinatorial representation theory and recent progress in signal processing on graphs, we investigate a novel, scalable transform method to interpret and exploit structure in ranked data. We represent data on the permutahedron using an overcomplete dictionary of atoms, each of which captures both smoothness information about the data (typically the focus of spectral graph decomposition methods in graph signal processing) and structural information about the data (typically the focus of symmetry decomposition methods from representation theory). These atoms have a more naturally interpretable structure than any known basis for signals on the permutahedron, and they form a Parseval frame, ensuring beneficial numerical properties such as energy preservation. We develop specialized algorithms and open software that take advantage of the symmetry and structure of the permutahedron to improve the scalability of the proposed method, making it more applicable to the high-dimensional ranked data found in applications.

Density order of Parseval wavelet frames from extension principles

We characterize approximation order and density order of those Parseval wavelet frames obtained from Oblique Extension Principle. These notions are closely related to approximation order and density order by a quasi-projection operator. To give our characterizations, we shall explain the behavior on a neighborhood of the origin of the Fourier transform of a refinable function. In particular, we invoke the classical notion of approximate continuity. We write our results in the multivariate context of Parseval wavelet frames associated to A, an expansive linear map preserving the integer lattice.

Characterization of wavelets associated with $AB$-MRA on $L^2(\mathbb R^n)$

A wavelet with composite dilations is a function generating an orthonormal basis or a Parseval frame for $L^2(\mathbb R^n)$ under the action of lattice translations and dilations by products of elements drawn from non-commuting matrix sets $A$ and $B$. Typically, the members of $B$ are matrices whose eigenvalues have magnitude one, while the members of $A$ are matrices expanding on a proper subspace of $\mathbb R^n$. In this paper, we provide the characterization of composite wavelets based on results of affine and quasi affine frames. Furthermore all the composite wavelets associated with $AB$-MRA on $L^2(\mathbb R^n)$ are also characterized.

New identity on Parseval p-approximate Schauder frames and applications

A very useful identity for Parseval frames for Hilbert spaces was obtained by Balan, Casazza, Edidin, and Kutyniok. In this paper, we obtain a similar identity for Parseval p-approximate Schauder frames for Banach spaces which admits a homogeneous semi-inner product in the sense of Lumer-Giles.

On Excesses and Duality in Woven Frames

Weaving frames in separable Hilbert spaces have been recently introduced by Bemrose et al. to deal with some problems in distributed signal processing and wireless sensor networks. In this paper, we study the notion of excess for woven frames and prove that any two frames in a separable Hilbert space that are woven have the same excess. We also show that every frame with a large class of duals is woven provided that its redundant elements have small enough norm. Also, we try to transfer the woven property from frames to their duals and vice versa. Finally, we look at which perturbations of dual frames preserve the woven property; moreover, it is shown that under some conditions, the canonical Parseval frame of two woven frames is also woven.

Perturbation and construction of almost self‐located robust frames with applications to erasure recovery

In the frame theory, stability is one of the most important properties of frames. In this paper, we investigate the stability of the almost self‐located robust frames, which can reconstruct signals from its unordered partial frame coefficients. We give conditions on the stability of almost self‐located robustness of frames in erasure recovery. We prove that the property of almost self‐located robustness is preserved with respect to an invertible operator, and consequently, every almost self‐robust frame can be transformed into an almost self‐located robust Parseval frame by using a special invertible transformation. We also present some construction methods of almost self‐located robust frames. In particular, we show that every frame can be rescaled to an almost self‐located frame after adding a suitable vector.

A novel approach on micropolar fluid flow in a porous channel with high mass transfer via wavelet frames

Abstract In this article, we present the Laguerre wavelet exact Parseval frame method (LWPM) for the two-dimensional flow of a rotating micropolar fluid in a porous channel with huge mass transfer. This flow is governed by highly nonlinear coupled partial differential equations (PDEs) are reduced to the nonlinear coupled ordinary differential equations (ODEs) using Berman's similarity transformation before being solved numerically by a Laguerre wavelet exact Parseval frame method. We also compared this work with the other methods in the literature available. Moreover, in the graphs of the velocity distribution and microrotation, we shown that the proposed scheme's solutions are more accurate and applicable than other existing methods in the literature. Numerical results explaining the effects of various physical parameters connected with the flow are discussed.

Lower Semi-frames, Frames, and Metric Operators

This paper deals with the possibility of transforming a weakly measurable function in a Hilbert space into a continuous frame by a metric operator, i.e., a strictly positive self-adjoint operator. A necessary condition is that the domain of the analysis operator associated with the function be dense. The study is done also with the help of the generalized frame operator associated with a weakly measurable function, which has better properties than the usual frame operator. A special attention is given to lower semi-frames: indeed, if the domain of the analysis operator is dense, then a lower semi-frame can be transformed into a Parseval frame with a (special) metric operator.

Hamiltonians Generated by Parseval Frames

It is known that self-adjoint Hamiltonians with purely discrete eigenvalues can be written as (infinite) linear combination of mutually orthogonal projectors with eigenvalues as coefficients of the expansion. The projectors are defined by the eigenvectors of the Hamiltonians. In some recent papers, this expansion has been extended to the case in which these eigenvectors form a Riesz basis or, more recently, a {mathcal{D}}-quasi basis (Bagarello and Bellomonte in J. Phys. A 50:145203, 2017, Bagarello et al. in J. Math. Phys. 59:033506, 2018), rather than an orthonormal basis. Here we discuss what can be done when these sets are replaced by Parseval frames. This interest is motivated by physical reasons, and in particular by the fact that the mathematical Hilbert space where the physical system is originally defined, contains sometimes also states which cannot really be occupied by the physical system itself. In particular, we show what changes in the spectrum of the observables, when going from orthonormal bases to Parseval frames. In this perspective we propose the notion of E-connection for observables. Several examples are discussed.

Full-spark frames arising from one-parameter groups

The present paper provides a procedure for constructing full-spark Parseval frames arising from the linear action of a 1-parameter group acting in . Precisely, given a square matrix A of order n, with real entries, we say that A induces the full spark frame property if the following conditions hold. There exists a vector v for which given any finite set of cardinality n, the collection is a basis for . First, we show that if the spectrum of A is not a subset of the reals, then A does not induce the full spark frame property. Secondly, we establish that if the spectrum of A is a subset of the reals, then A induces the full spark frame property if and only if every eigenvalue of A has geometric multiplicity one. The proof of the latter fact gives a new algorithm for the construction of a class of full spark Parseval frames for .

Approximation by group invariant subspaces

In this article we study the structure of Γ-invariant spaces of L2(S). Here S is a second countable LCA group. The invariance is with respect to the action of Γ, a non commutative group in the form of a semidirect product of a discrete cocompact subgroup of S and a group of automorphisms. This class includes in particular most of the crystallographic groups. We obtain a complete characterization of Γ-invariant subspaces in terms of range functions associated to shift-invariant spaces. We also define a new notion of range function adapted to the Γ-invariance and construct Parseval frames of orbits of some elements in the subspace, under the group action. These results are then applied to prove the existence and construction of a Γ-invariant subspace that best approximates a set of functional data in L2(S). This is very relevant in applications since in the euclidean case, Γ-invariant subspaces are invariant under rigid movements, a very sought feature in models for signal processing.

Model subspaces techniques to study Fourier expansions in L2 spaces associated to singular measures

Let μ be a probability measure on T that is singular with respect to the Haar measure. In this paper we study Fourier expansions in L2(T,μ) using techniques from the theory of model subspaces of the Hardy space. Since the sequence of monomials {zn}n∈N is effective in L2(T,μ), it has a Parseval frame associated via the Kaczmarz algorithm. Our first main goal is to identify the aforementioned frame with boundary values of the frame Pφ(zn) for the model subspace H(φ)=H2⊖φH2, where Pφ is the orthogonal projection from the Hardy space H2 onto H(φ). The study of Fourier expansions in L2(T,μ) also leads to consider positive kernels in the Hardy space. Our second main goal is to study the set of measures μ which reproduce a kernel contained in a model subspace. We completely characterize this set when the kernel is the reproducing kernel of a model subspace, and we study the consequences of this characterization.

Parseval Proximal Neural Networks

The aim of this paper is twofold. First, we show that a certain concatenation of a proximity operator with an affine operator is again a proximity operator on a suitable Hilbert space. Second, we use our findings to establish so-called proximal neural networks (PNNs) and stable tight frame proximal neural networks. Let mathcal {H} and mathcal {K} be real Hilbert spaces, b in mathcal {K} and T in mathcal {B} (mathcal {H},mathcal {K}) a linear operator with closed range and Moore–Penrose inverse T^dagger . Based on the well-known characterization of proximity operators by Moreau, we prove that for any proximity operator mathrm {Prox}:mathcal {K}rightarrow mathcal {K} the operator T^dagger , mathrm {Prox}( T cdot + b) is a proximity operator on mathcal {H} equipped with a suitable norm. In particular, it follows for the frequently applied soft shrinkage operator mathrm {Prox}= S_{lambda }:ell _2 rightarrow ell _2 and any frame analysis operator T:mathcal {H}rightarrow ell _2 that the frame shrinkage operator T^dagger , S_lambda , T is a proximity operator on a suitable Hilbert space. The concatenation of proximity operators on mathbb R^d equipped with different norms establishes a PNN. If the network arises from tight frame analysis or synthesis operators, then it forms an averaged operator. In particular, it has Lipschitz constant 1 and belongs to the class of so-called Lipschitz networks, which were recently applied to defend against adversarial attacks. Moreover, due to its averaging property, PNNs can be used within so-called Plug-and-Play algorithms with convergence guarantee. In case of Parseval frames, we call the networks Parseval proximal neural networks (PPNNs). Then, the involved linear operators are in a Stiefel manifold and corresponding minimization methods can be applied for training of such networks. Finally, some proof-of-the concept examples demonstrate the performance of PPNNs.

Laguerre Wavelets Exact Parseval Frame-based Numerical Method for the Solution of System of Differential Equations

In this study, the Laguerre wavelets exact Parseval frame is introduced and proposed an effective numerical algorithm to get a numerical solution for the system of differential equations based on the Laguerre wavelets exact Parseval frame. This algorithm includes the collocation method and truncated Laguerre wavelet frames. Here, we reduce the system of differential equations into a set of algebraic equations which are having unknown Laguerre wavelet frame coefficients. Some numerical examples are given and compared to the numerical solution by the present method with the Adomian decomposition method. Moreover, the modeling of the spreading of a non-fatal disease in a population, which represents a system of an ordinary differential equation is numerically solved by the proposed technique and compared with the Adomina decomposed method. The obtained results reveal that the present algorithm provides a good approximation than existing methods.